Study of the Proper Generalized Decomposition for real-time applications
Document typeBachelor thesis
Rights accessRestricted access - author's decision
Computational simulations are valuable tools for engineers because they facilitate design processes. Complex systems or problems are unreachable by the standard methods (like Finite Element Method, FEM) and Model Order Reduction (MOR) techniques are required. Among those, the Proper Orthogonal Decomposition has proved to be useful in many fields of science and engineering, and especially in solid mechanics. Alternatively, an a priori strategy MOR, the Proper Generalized Decomposition (PGD), is being developed to solve parametric problems in what, in certain areas, are considered "real-time" conditions. The purpose of this study is to introduce the PGD emergent technique and its potential. This approach has been put into context by qualitatively describing MOR techniques when applied in computational solid mechanics, like the also introduced FEM. The PGD key features and advantages, as well as drawbacks, have been introduced in comparison with the POD, a widely extended technique. Regarding the potential of the PGD, its adequacy in augmented reality in computational surgery has already been documented. In this study, the PGD has first been implemented to solve two 2D academic problems successfully (Poisson equation and linear elastics) with full mathematical formulation. Next, based on the linear elastic case study, a real-time application program has been contrived. In the latter, the problem’s configuration has been updated, adding more complexity. Throughout this report, the good results obtained with the PGD have been demonstrated. Results show, first, the ability to solve simple cases using a PGD code applied in Matlab and, therefore, validates this approach. Second, it has been proved that it is useful to develop real-time applications, performing as expected. However, there are still open concepts and related issues to improve the methodology. The study has achieved to develop a simple and comfortable approach to the PGD. At the same time, it provides the readers with the basic tools to encourage them to make their first steps in the PGD methodology. With the hope that all this will allow the reader understand the importance and potential of this methodology.
1) Understand the basic principles of the so-called "Proper Generalized Decomposition" (PGD) for the solution of multidimensional and multiparameter boundary value problems2) Develop a Matlab program for addressing the solution of Heat Conduction and Elastostatic problems in 2D using the PGD.3) Apply the Matlab code in a real-time application, such as the fat prediction of displacements and reaction forces of a structure under an arbitrarity placed load.
SubjectsNumerical analysis, Finite element method, Decomposition method, Anàlisi numèrica, Elements finits, Mètode dels, Descomposició, Mètode de
DegreeGRAU EN ENGINYERIA EN TECNOLOGIES AEROESPACIALS (Pla 2010)
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